Crossover from Bose-Einstein Condensed Molecules to

Cooper Pairs by Using Feshbach Resonance

Shu-Wei Chang

Department of Electrical and Computer Engineering,

University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA

(Dated: May 6, 2004)

Abstract

At low temperature, fermionic atoms can either condense into a Bose-Einstein condensate (BEC)

by forming bosonic molecules or be loosely paired to form Cooper pairs described by Bardeen,

Cooper, and Schrieffer’s (BCS) microscopic theory. Experimentalists have claimed the observation

of the intermediate regime between BEC and BCS limits by tuning the scattering length with the

aid of Feshbach resonance. In this report, we will discuss the idea of using Feshbach resonance

to achieve the crossover. Some many-body formulation concerning this crossover will be briefly

introduced, but not in detail. The experimental results will be our main concerns and addressed

in this report.

1

I. INTRODUCTION

After the successful demonstrations of Bose-Einstein condensation (BEC) of atomic

gas,[1, 2], the cooling down of atomic Fermi gas has also been achieved recently.[3, 4] Unlike

bosonic atoms, fermionic atoms cannot condense into a single state due to anti symmetriza-

tion of the macroscopic wave function. However, two fermionic atoms can be converted into

a bosonic molecule and condense into a BEC again.

Whether two fermionic atoms can form a bosonic molecule is closely related to the two-

body physics of the atoms. The sign of the s-wave scattering length between two identical

fermionic atoms in different internal quantum states, called an open channel, determines

whether the effective interaction in that channel between two fermions is repulsive or attrac-

tive at low temperature.[5] The sign of the s-wave scattering length has a close connection

with the near-by energy of the bound state in a closed channel of which threshold energy is

higher than that of the open channel. For fermionic atoms with the same internal quantum

states, the s-wave scattering cross section disappears due to anti-symmetrization of wave

functions, and one has to consider scattering lengths due to, for example, p-wave scatter-

ing. Experimentally, it will be more efficient to observe the change of the s-wave scattering

length in an open channel characterized by the scattering of the atoms with different internal

quantum numbers.[6, 7] However, if the physical environment favors the atoms of a single

internal quantum state, issues on p-wave scattering such as the analogy of p-wave pairing

in superconductivity then can be discussed.[8]

For atoms, the s-wave scattering length can be controlled by mechanisms such as the

interaction of magnetic moments with magnetic field[9] or Raman transition induced by

lasers.[10, 11] In either case, the energy of the bound state in the closed channel is swept

across the threshold energy of the open channel. It is possible to change the sign of the

scattering length and thus the property of the effective interaction. For fermionic atoms,

the controllability of the scattering length enables the formation of a correlated atom pair in

one limit and a tightly-bound molecule in the other, as will be discussed in the next section.

Experimentally, it is interesting to ask what happens to the pair of fermionic atoms at the

crossover between the two limits. Theoretically, it can also provide an analogue test for one

of the candidate theories of high temperature superconductivity, originated from Nozières

and Schmitt-Rink by considering electron pairs resonant with the molecular state.[12]

2

Eb

E

Eth

Close Channel

Open Channel

Atomic Separation

FIG. 1: Feshbach resonance of a quasi-bound state with the open channel. Eb is the resonant

energy of the quasi-bound state. Eth is the threshold energy of the open channel. The blue part

indicates the continuum of the open channel.

II. FESHBACH RESONANCE

Feshbach is first discussed in nuclear physics.[13] In this section, emphasis will be put on

the Feshbach resonance using magnetic field. More details can be found in Ref. [5].

For fermionic atoms, Feshbach resonance is the change of the scattering length in an open

channel due to the coupling to a bound state in the closed channel. The state in the closed

channel will be a true bound state if either there is no coupling to the open channel, or the

energy of the state lies below the threshold energy of the open channel. On the other hand,

if the state has an energy lying in the continuum of the open channel and exhibits a finite

coupling to the open channel, it should be called a quasi-bound state since it is not a true

bound state. A quasi-bound state exhibits some sort of energy resonance just as a bound

state but is characterized by a finite decay width due to the coupling to the open channel.

Fig. 1 shows the Feshbach resonance of a quasi-bound state coupling to the open channel.

Write the general wave function |Ψ > as a superposition of the part in the open chan-

nel |Ψ1 > and the part in the closed channel |Ψ2 >. The two parts are orthogonal to each

other: < Ψ2|Ψ1 >= 0. The Hamiltonian describing both open and closed channels is de-

noted as H. Also, write the projection operator to the open and closed channels as P1 and

P2. We define various projected Hamiltonians as

Hij = PiHPj, i, j = 1, 2 (1)

3

After a close look, the effective Hamiltonian Heff describing the open channel is written as

Heff |Ψ1 >=[

H11 + H12(E − H22 + iδ)−1H21

]

|Ψ1 >= E|Ψ1 > (2)

where δ is a small number. We further write the projected Hamiltonian H11 as H11 = H0+U

where H0 is the noninteracting part of the Hamiltonian H11. The interaction term Ueff in

the effective Hamiltonian Heff is

Ueff = U + H12(E − H22 + iδ)−1H21 (3)

The effective s-wave scattering length is evaluated from equation (3) by Born’s approxima-

tion for the zero momentum state |ϕ0 > of the open channel. The energy of the state |ϕ0 >

is just the threshold energy Eth of the open channel. Denote a complete set of the eigenstates

of the projected Hamiltonian H22 as {|φn >, n = 1, 2, ...} with eigenenergies {En}. Due to

the fact that H12 = H†21 and the completeness relation 1 =

∑

n |φn >< φn| in the closed

channel, the s-wave scattering length as of this open channel can be written as

as =mr

4π~2< ϕ0|Ueff |ϕ0 >

=

as,1︷ ︸︸ ︷mr

4π~2< ϕ0|U |ϕ0 > +

mr4π~2

∑

n

| < φn|H21|ϕ0 > |2

Eth − En + iδ

=

[

as,1 +mr

4π~2

∑

n,En 6=Eb

| < φn|H21|ϕ0 > |2

Eth − En + iδ

]

︸ ︷︷ ︸

anr

+mr

4π~2| < φb|H21|ϕ0 > |

2

Eth − Eb + iδ(4)

where as,1 denotes the s-wave scattering length in the open channel; |φb > is the (quasi-)

bound state with energy Eb nearest to the threshold energy Eth; and ans is the nonresonant

part of the s-wave scattering length. The contribution from the state |φb > is separated from

the summation since it is the most significant. The denominator Eth −Eb in equation (4) is

a function of the applied magnetic field B. Denote the internal quantum numbers labeling

the two atoms in the open channel as α and β where α 6= β. Near a particular magnetic

field B0, the denominator can be expanded as

Eth − Eb ≈ (µb − µα − µβ)(B − B0) (5)

where µb, µα, and µβ are the magnetic moments of the (quasi-) bound state and the states la-

beled by internal quantum numbers α and β. If the nonresonant s-wave scattering length anr

does not vary too much with the magnetic field, equation (4) can be written as

4

B

as

anr

B0

FIG. 2: S-wave scattering length as a function of magnetic field.

as = anr(1 +∆B

B − B0 + iδB)

∆B =mr

4π~2anr

| < φb|H21|ϕ0 > |2

µb − µα − µβ(6)

where δB is an infinitesimal quantity. A typical s-wave scattering length as a function of

the magnetic field is shown in Fig. 2 for anr > 0 and ∆ − B > 0. The scattering length

diverges to plus and minus infinities as the magnetic field tends to B−0 and B+0 . Examples

include the s-wave scattering length for |F = 9/2,mF = −9/2 > +|F = 9/2,mF = −7/2 >

of 40K[6, 14, 15] and so on. Also, this scattering is an elastic one since the kinetic energy is

not lost during the scattering.

If the threshold energy Eth is slightly larger than Eb, the s-wave scattering length is

positive. Although the effective interaction potential corresponding to this positive elastic

scattering length is repulsive, two fermionic atoms can form a molecule when the magnetic

field is swept downward across the magnetic field B0. In this case, it is possible to form

BEC from these bosonic molecules. The energy difference between the two atoms and

molecule is brought away by three-body processes instead of the s-wave two-body inelastic

scattering which is believed to be absent.[8, 16–19] However, the molecules produced in this

way are usually highly vibrationally excited. The deexcitation of these molecules will cause

significant inelastic scattering, and the loss rate of these molecules is high.[6, 7, 20] On the

other hand, for the negative scattering length (attractive interaction), atoms in the open

channel cannot form a stable molecule described by a true bound state. They can interact

via the quasi-bound state and form a correlated pair, or may be viewed as molecules with

finite lifetime. This regime should be described by a generalized Bardeen, Cooper, and

Schrieffer’s (BCS) theory instead of BEC. Typically, the atoms in this regime have lower

loss rate than the molecules in the other side of the resonance. [6, 7, 20]

5

III. ON THE MANY-BODY ASPECTS

In atomic gases, the presence of quasi-bound state is easily modeled by adding a coupling

term to a molecular state in the BCS Hamiltonian for fermionic atoms.[21–23]

H =∑

k

²k(a†k,↑ak,↑ + a

†k,↓ak,↓) + Ubg

∑

k1+...+k4=0

a†k1,↑a†k2,↓

ak3,↓ak4,↑

+g∑

k,q

(b†qaq/2+k,↑aq/2−k,↓ + bqa†

q/2−k,↓a†

q/2+k,↑) +∑

q

(ν + Eq)b†qbq (7)

where a (b) and a† (b†) are the annihilation and creation operators of atoms (molecules); ²k

and Eq are the the kinetic energies of atoms and molecules; ↑ and ↓ are pseudo spins indi-

cating the two different internal quantum states; Ubg is the background attractive potential

between atoms; µ is the bound-state energy relative to the threshold of the open channel

due to the magnetic field; and g describes the exchange between atoms and molecules. This

Hamiltonian describes the superfluidity of Fermi atoms in the BEC limit, BCS limit, and

the intermediate regime. The change of the scattering length is not obvious in equation (7)

but is implicitly included due to the coupling constant g. This model is a more rigorous

picture than the two-body one in the previous section because the corresponding effective

theory fails if the scattering length is large near the resonance.

Instead of going through the mathematical details, we only mention a few results of key

calculations. The detailed treatments of this Hamiltonian or its mutants for atomic Fermi

gas can be found in literatures.[21–25]

Fig. 3 shows the critical temperature, the chemical potential, and the fractions of atoms

and molecules at the emergence of superfluidity as a function of the bound-state energy.[23]

This result is obtained numerically by self-consistently solving the gap equation and the

conservation of particle number.

1 = (−Ubg +g2

ν − 2µ)∑

k

tanh[β(k2/2m − µ)/2]

2(k2/2m − µ)

N = 2∑

k

1

eβ(k2/2m−µ) + 1+ 2

∑

k

1

eβ(k2/4m+ν−µ) + 1−

1

β

∑

q

∂

∂µln[1 − χ(q, iω)] (8)

where µ is the chemical potential; N is the total particle number; and χ(q, iω) is a correction

function describing the number fluctuations of atoms and molecules. In Fig. 3(a), the

solid line shows the critical temperature calculated from equation (8). The dash line is

a calculation by substituting equation (6) into Ubg and solving the BCS gap equation. The

6

(a) (b) (c)

FIG. 3: Physical quantities at the emergence of superfluidity as a function of the bound-state

energy. (a) Critical temperature. Solid line is the result from equation (8). Dash-line is the result

from the reduced BCS theory. (b) Chemical Potential. (c) Fractions of atoms and molecules. Solid

line is the fraction of the fermions while the dot-dash line is the pair fraction. After Ref. [23]

critical temperature calculated from simple BCS gap equation diverges at the resonance

because the interaction energy Ubg calculated in this way diverges to minus infinity. The

solid line matches well with the simple BCS theory when the bound-state energy is above

the threshold. On the other hand, it approaches the critical temperature predicted by BEC

theory at the other limit. As the bound-state energy is tuned downward from the BCS side

toward the resonance, the critical temperature increases, which makes this model a candidate

for high-temperature superconductivity. There is an optimized critical temperature close

to the threshold energy. This critical temperature is around 0.26 TF , where TF is the

Fermi temperature of the atoms without any molecules. Similarly in Fig. 3(b), the chemical

potential is also a smooth connection between the two limits. In the BCS limit, the chemical

potential approaches that of the Fermi gas at critical temperature. In the BEC limit, the

chemical potential will tend to one half of the bound-state energy. This is an analogy to

that the chemical potential of pure bosons approach their ground energy as BEC occurs.

Fig. 3(c) shows the fractions of fermionic atoms and atom pairs. The fraction of fermions

approach zero as the system goes deep into BEC limit while the pair fraction shows the

opposite trend. The atom pairs resemble Cooper pairs above the resonance but are like

molecules below the resonance.

The calculation stated above is only related to the physical quantities when the super-

fluidity phase occurs. The question on whether there is a transition between BEC and

7

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

T/T

F

B-B0 (Gauss)

BEC BCS

TF=0.35 µK

(a)

0.1

0.15

0.2

00.2

0.40.6

0

0.03

0.06

T/TFB−B

0 (Gauss)

nm

c/n

m

(b)

FIG. 4: (a) The phase diagram of BCS-BEC transition. The upper dash line is the critical temper-

ature calculated from simple BCS theory. The lower dash line is the BCS-BEC boundary. (b) The

fraction of BEC condensate as a function of the magnetic field and temperature. After Ref. [26]

BCS limits, i.e., a boundary between BEC and BCS regimes in the phase diagram shown in

Fig. 4(a), is also interesting.[26] At zero temperature, write the pair wave function as the

superposition of the open and closed-channel wave functions χm(x, x′) and χaa(x, x

′),

√

Z(B)χm(x, x′)|closed > +

√

1 − Z(B)χaa(x, x′)|open > (9)

the question of BCS-BEC crossover becomes whether we can find a critical magnetic field

Bc(T = 0) such that the parameter Z(B < Bc(T = 0)) = 1 and Z(B > Bc(T = 0)) = 0. At

finite temperatures less than the critical temperature, the question becomes whether there is

a critical magnetic field Bc(T ) below which molecules begin to condense into the molecular

ground state, i.e., nm,BEC(B < Bc(T ), T )/nm(B < Bc(T ), T ) > 0. We first note that the

critical temperature calculated in Ref. [23] has a different shape from that calculated in

Ref. [26]. It may be a result of different approximations used in the calculations. Fig. 4(b)

shows the BEC fraction as a fraction of the magnetic field and temperature. At a given

temperature, there is a critical magnetic field Bc(T ) where the BEC condensate emerges.

These critical magnetic fields Bc(T ) deviate from B0 predicted by simple two-body physics.

A simple argument about these deviations is that molecules, though with a finite lifetime in

the BCS regime, will emerge from atomic Fermi sea when the shifted ground state energy

of molecules due to the coupling is roughly twice the chemical potential of the system.

This early formation of the molecular condensate before the exact resonance makes the

superfluidity at this crossover unusual.

8

(a)

3

2

1

0

N -

2 [10

-9]

6004002000Time [ms]

60

40

20

0

N -

1 [10

-6]

a)

b)

(b)

FIG. 5: (a) The scattering length of 40K. (b) The inelastic scattering rate of 6Li. The upper and

lower figure are the inverse and inverse square of the population as a function of time, indicating

the two-body and three-body processes. After Ref. [9, 14].

IV. EXPERIMENTS ON BCS-BEC CROSSOVER

Most of the experiments on BCS-BEC crossing are aimed at fermionic atoms 6Li and

40K.[2, 4, 6, 7, 9, 14, 15, 17–19] For 6Li, two hyperfine states |1/2, 1/2 > and |1/2,−1/2 >

are used while for 40K, two states |9/2,−7/2 > and |9/2,−9/2 > are used.

Fig. 5(a) shows the elastic scattering length of 40K as a function of magnetic field.[14]

The Feshbach resonance of 40K takes place at 224.18 G. The magnitude of the scattering

length is obtained from the extraction of rethermalization time by heating the atomic gas

when modulating the power of the optical trap.[9] The sign of the scattering length is

obtained by measuring the mean-filed shift of the transition |9/2,−9/2 >−→ |9/2,−5/2 >

.[15] The change of the scattering length is very significant as the magnetic field sweeps

across the resonance frequency, indicating the ability to control the effective interaction

between 40K atoms. Fig. 5(b) shows the inelastic loss rate of 6Li at 680 G.[17] There are two

Feshbach resonance peaks for 6Li–550 G and 822 G. The resonance at 822 G has a very broad

lineshape, and thus the inelastic scattering at 680 G is believed to be greatly influenced by

this resonance. The upper plot of Fig. 5(b) shows the inverse of the number of particles

remaining in the trap as a function of time. The nearly linear behavior seems to imply

that the two-body inelastic process dominates in the time scale observed. The lower plot

shows the inverse square of the population as a function of time. The three-body process will

dominate over the two-body process only when the population is high. The small linear part

at time close to zero thus indicates the three-body process. However, the inelastic two-body

9

201 202 2030

20

40

60

80

100re

main

ing

ato

ms

(%)

B (gauss)

(a)

Ato

mic

Fra

ctio

n

Magnetic Field [G]

1.0

0.5

0.0

900850800750

(b)

FIG. 6: (a) The fraction of 40K atoms as a function of the magnetic field. (b) The fraction of 6Li

atoms as a function of the magnetic field. Cross: initial ramp-down speed=30 G/µs; circle: linear

ramp-down at 100 G/ms; triangle: linear ramp-down at12.5 G/ms. After Ref. [7, 19].

loss is believed to be suppressed. In Ref. [17], the author thus attributed the observed loss

rate to the three-body process simultaneously accompanying by the temperature change.

Fig. 6(a) and (b) show the fractions of fermionic atoms 40K at 224.18 G[19] and 6Li at

822 G[7] as a function of the magnetic field. In either case, the measurement begins from

the BEC side where bound molecules exist. The magnetic field is first swept up to a given

value. If the given magnetic field exceeds the resonant magnetic field, the molecules will

dissociate. During the sweeping which takes about more than ten milliseconds, the molecular

or atomic gas will expand so that the density of the gas is diluted. This action is performed

to avoid the unwanted interference from many-body effects and the reformation of molecules

at successive stage.[6, 7] After the expansion, the magnetic field is fast ramped down to zero.

The fraction of the atoms is then obtained at zero field by optical image method which only

detects the atoms because the transition of molecules does not match the frequency of the

incident light.[6, 14, 27] From Fig. 6(a) and (b), the fraction of atoms greatly increase as the

given magnetic field goes above the resonance, indicating the absence of molecules. Also, in

Fig. 6(b), the fraction of atoms corresponding to different speeds of ramping the magnetic

field down to zero are shown. This ramp-down speed has to be carefully chosen because the

resonant field B0 obtained in this way will shift due to the ramp-down speed.[7] However,

the ramp-down speeds used in the experiments do not significantly change the resonant field.

Fig. 7(a) and (b) show the fraction of the condensate of pair atoms (This term should

10

-0.5 0 0.5 1.00

0.05

0.10

0.15

-9 -6 -3 0 3 16 18198

200

202

204

B(g

auss

)

t (ms)N

0/N

∆B (gauss)

(a)

Co

nd

en

sa

te F

ractio

n

Magnetic Field [G]

0.8

0.4

0.0

1000800600

(b)

FIG. 7: (a) The decay of 40K2 condensate fraction. Circle: 2 ms hold time; triangle: 30 ms hold

time (b) The decay of 6Li2 condensate fraction. Cross: 2 ms hold time; square: 100 ms hold time;

circle: 10 s hold time. After Ref. [6, 7].

stand for BEC molecules in the BCS side although its real meaning is a bit ambiguous in

these papers.) as a function of magnetic field for 40K[6] and 6Li.[7] Unlike Fig. 6(a) and (b),

the initial magnetic field is set at the BCS side and than brought slowly to a given magnetic

field to avoid the enhanced loss out of the optical trap due to rapid sweeping. No expansion

of atomic gas takes place at this stage. Furthermore, sometimes it is even necessary to in

crease optical power to enhance the confinement of the atomic gas.[7] The atomic gas is

held there for a period of time which is variable in the experiments. After the hold period,

the trap is turned off to allow the gas to expand. Simultaneously[6] or immediately after

the expansion,[7] the magnetic field is soon turned off. Unlike the experiments mentioned

previously, the atom in the gas will be adiabatically turned into molecules due to high

density. The fraction of molecular BEC condensate, which is assumed to be invariant before

and after the turn-off by some subtle argument, is then measured by mapping out the zero

momentum component[7] or Thomas-Fermi profile of the molecules.[6]

In Fig 7 (a) and (b), the fraction of the condensate decays as the hold time increases. As

mentioned in section II, the formation of molecules by sweeping the magnetic field across the

resonance creates vibrationlly-excited molecules. The deexcitation of these molecules will

cause the inelastic scattering and quench the fraction od the BEC condensate. The longer

the hold time is, the more thorough the quenching is. It accounts why significant decay is

observed at longer hold time. Close to the resonance and in the BCS side, the condensate

decays slowly while it decays faster farther away from the resonance in the BEC side because

the molecules are more tightly-bound. The extra bound energy has to be converted into

11

0

0.05

0.10

0.15

0.20

-0.4-0.2

00.2

0.40.6

0.8

-0.05

0

0.05

0.10

0.15

N /N0

T/TF

∆B (gauss)

(a)

0

0.2

0.4

0.6

Condensate

Fra

ction a

t 820 G

1000900800700Magnetic Field [G]

0.5

0.3

0.1

0.7

0.8

0.2

0.05

T/TF

(b)

FIG. 8: The fractions of (a) 40K2 BEC condensate and (b)6Li2 BEC condensate as a function of

the magnetic field and temperature. After Ref. [6, 7].

kinetic energy and reduces the condensate fraction.[28]

The argument that the condensate fraction measured in this way is just the condensate

fraction before the turn-off of the magnetic field is subtle. This argument will lead to the

conjecture that the BEC is present in the BCS side.[26] The argument is based on two

assumptions. The first one is that no BEC condensate is created when the magnetic field

drops to zero.[14] The second one is that slightly-correlated atoms tend to randomly pick up

a neighboring atom to form a molecule. Molecules formed in this way have finite momenta

and do not belong to the condensate. Therefore, the original condensate will be directly

projected to the condensate at zero magnetic field.[7] The condensate fraction as a function

of magnetic field and temperature for a given number of atoms is also measured.[6, 7] Fig. 8

(a) and (b) show the corresponding three-dimensional and contour plots for 40K2 and6Li2

molecules. Fig. 8 (a) can be directly compared with Fig. 4 (b) since this figure is an attempt

to fit the experimental data.

V. CONCLUSION

We have briefly surveyed the recent experiments on BEC-BCS crossover. Instead of going

into complicated calculations of many-body dynamics, we only look into those related to

thermal equilibrium. The interesting point of the cold fermionic atoms is the capability of

tuning the scattering length by magnetic field. BEC and BCS types of superfluidity can

thus be achieved in the same system by tuning this controllable parameter. Further, an

interesting regime of superfluidity between BEC and BCS has been opened. More future

work has to be devoted to explore the properties of the superfluidity phase in the regime.

12

[1] M. H. Anderson et al., Science 269, 198 (1995).

[2] K. B. Davis et al., Phys. Rev. Lett. 75, 3969 (1995).

[3] F. A. van Abeelen, B. J. Verhaar, and A. J. Moerdijk, Phys. Rev. A 55, 4377 (1997).

[4] B. DeMarco and D. S. Jin, Science 285, 1703 (1999).

[5] C. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University

Press, Cambridge, UK, 2002), 1st ed.

[6] C. Regal, M. Greiner, and D. Jin, Phys. Rev. Lett. 92, 040403 (2004).

[7] M. W. Zwierlein et al., Phys. Rev. Lett. 92, 120403 (2004).

[8] J. L. Bohn, Phys. Rev. A 61, 053409 (1999).

[9] T. Loftus et al., Phys. Rev. Lett. 88, 173201 (2002).

[10] F. K. Fatemi, K. M. Jones, and P. D. Lett, Phys. Rev. Lett. 85, 4462 (2000).

[11] S. Kokkelmans, H. M. J. Vissers, and B. J. Verhaar, Phys. Rev. Lett. 63, 031601(R) (2001).

[12] P. Nozières and S. Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985).

[13] H. Feshbach, Ann. Phys. 19, 287 (1962).

[14] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Nature (London) 424, 47 (2003).

[15] C. A. Regal and D. S. Jin, Phys. Rev. Lett. 90, 230404 (2003).

[16] V. A. Yurovsky and A. Ben-Reuven, Phys. Rev. A 60, R765 (1999).

[17] K. Dieckmann et al., Phys. Rev. Lett. 89, 203201 (2002).

[18] C. A. Regal, C. Ticknor, J. L. Bohn, and D. S. Jin, Phys. rev. Lett. 90, 053201 (2003).

[19] C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 083201 (2004).

[20] K. E. Strecker, G. B. Partridge, and R. G. Hulet, Phys. Rev. Lett. 91, 080406 (2003).

[21] M. Holland et al., Phys. Rev. Lett. 87, 120406 (2001).

[22] Y. Ohashi and A. Griffin, Phys. Rev. Lett. 89, 130402 (2002).

[23] J. N. Milstein, S. Kokkelmans, and M. J. Holland, Phys. Rev. A 66, 043604 (2002).

[24] M. Mackie, K.-A. Suominen, and J. Javanainen, Phys. Rev. Lett. 89, 180403 (2002).

[25] P. Naidon, Grancoise, and Masnou-Seeuws, Phys. Rev. A 68, 033612 (2003).

[26] G. M. Falco and H. T. C. Stoof, Phys. Rev. Lett. 92, 130401 (2004).

[27] M. W. Zwierlein et al., Phys. Rev. Lett. 91, 250401 (2003).

[28] J. Cubizolles et al., Phys. Rev. Lett. 91, 240401 (2003).

13